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2005-07-19

Cutting and Repairing Diagonal Lines

David Bell recently asked if it was possible to detect a long diagonal line without
destroying it, and if it was possible to send a signal through a diagonal line. The answers
to both questions hinge on the ability to cleanly break and then repair the line. The
glider constructions for such actions, while not optimum, have been found.

Bell showed that it was possible to repair a two cell break with a Loaf predecessor,
and then Karel Suhajda and H.Koenig were able to find a 3 Glider construction of a Loaf that
worked perfectly. Mark Niemiec also posted some similar contructions by David Buckingham which can
also be used to close up a line.

A way to break a line was then posted by H.Koenig. Two gliders cause the diagonal line
to become a pair of clean burning fuses, and then a pair of reactions consisting of two Gliders
and a Lightweight Spacehip (LWSS) (previously discovered by Jason Summers) create the domino
sparks which halts the burning fuses. This
results in a twenty bit gap where the diagonal line used to be. This construction is
not complete, however, because two of the gliders would interfere with each other if
they came from infinity. Also, the gap created is much larger than it could be.

David Greene then showed how to use a pair of reactions using a Pond, a LWSS and
three Gliders to lengthen a diagonal line by 4 bits can be used to close a gap by 8 bits. So
a gap of 8n+2, with 10 bits as the minimum, becomes the more useful reactions.

Bell then showed how a Glider collision with a stable object could be used to cleanly
cut a line. The added cost in Gliders used to construct and place the stable object
(in this example, a Block and a Loaf) is outweighed by the flexibility in timing and
and in the placement of the reaction which ends the burning of the fuse. Greene also showed a
a way to place and ignite a Tub to make a 10 cell gap. But the 9 cell gap can be expanded
to 10 by simply delaying one of the fuse stablizers by one generation and shifting it slightly.
(Any reaction can be made arbitrarily wide in the same fashion.)

All of these reactions require a glider to be placed as close as possible to the diagonal line.
Bell showed an example of a reaction of 2 LWSSs and a Glider which can place the
Gliders needed in the previous reactions, as well as one that can place a Loaf near the line
for the 9 bit gap.

There are some issues remaining. It would be useful if all the Gliders and LWSSs in the gap cloing reactions
camefrom the same side of the diagonal line, as that would make timing issues a lot easier. Also
a demonstration of a reaction that can detect the presence of the line needs to be made. Finally,
for use
in other patterns, the reactions which cut or repair a diagonal line that are triggered
by a single Glider (or a set of Herschel Track components) needs to be actually built.